Question

A line passes through (9, –9) and (10, –5).
a. Write an equation for the line in point-slope form.
b. Rewrite the equation in standard form using integers.


y + 9 = 4(x – 9); –4x + y = –45


y – 9 = 4(x + 9); –4x + y = 45


y + 9 = 4(x + 9); –4x + y = –45


y – 9 = 4(x – 9); –4x + y = 45

Answer 1

To find the equation of the line that passes through the points \((9, -9)\) and \((10, -5)\), we will start by calculating the slope of the line.

### Step 1: Calculate the slope
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points \((9, -9)\) and \((10, -5)\):

- \( x_1 = 9\), \( y_1 = -9 \)
- \( x_2 = 10\), \( y_2 = -5 \)

So,

\[
m = \frac{-5 - (-9)}{10 - 9} = \frac{-5 + 9}{10 - 9} = \frac{4}{1} = 4
\]

### Step 2: Write the equation in point-slope form
The point-slope form of the equation of a line is:

\[
y - y_1 = m(x - x_1)
\]

Using the point \((9, -9)\) and the slope \( m = 4\), we can write:

\[
y - (-9) = 4(x - 9)
\]

Which simplifies to:

\[
y + 9 = 4(x - 9)
\]

### Step 3: Convert to standard form
To convert the point-slope form to standard form \(Ax + By = C\), we start with the equation we derived:

\[
y + 9 = 4(x - 9)
\]

Expanding the right side:

\[
y + 9 = 4x - 36
\]

Now, subtract \(4x\) from both sides and rearrange the equation:

\[
-4x + y + 9 = -36
\]

Subtract \(9\) from both sides:

\[
-4x + y = -45
\]

Finally, we can multiply through by \(-1\) to make the \(A\) coefficient positive (which is generally desired in standard form):

\[
4x - y = 45
\]

### Summary
a. The equation in point-slope form is:
\[
y + 9 = 4(x - 9)
\]

b. The equation in standard form using integers is:
\[
4x - y = 45
\]

From the options you provided:

The correct statement is: **y + 9 = 4(x - 9); –4x + y = 45**.